My subject is optimal control of PDE and I want to put optimal control problems in Matlab to be solved but I don't know how to do this. I would be grateful if you could give me some references in that direction.
An example of what I want is the following one: Consider the problem:
$$\begin{cases} \dfrac{\partial{y}}{\partial{t}} -\Delta y(t,x_1,x_2,x_3)=u(t)\cdot f(t,x_1,x_2,x_3) \\ y(t,x_1,x_2,x_3)=0, \ \forall\ t\in [0;T],\ \forall x=(x_1,x_2,x_3)\ \text{with}\ ||x||=1 \\ y(0;x_1;x_2;x_3)=g(x_1,x_2,x_3)\end{cases}$$
where $f,g$ are given functions and $u:[0;\infty]\to [0;1]$ is a control in $L^2$.
I want to find the control $u$ that maximizez the folowing cost functional:
$$I(u)=\int_{0}^{T}\int_{B(0;1)} |y(t,x_1,x_2,x_3)| dx dt$$.
I know that for given $u$ i can solve the PDE with matlab tools or via a free element method (with FreeFem), but I do not see how write this problem in Matlab or other softwares. Any reference would be very appreciated by me. My goal is to simulate in a software the problems that I study in optimal control PDE.
Is there a software that solves numerically optimal control problems, even for ODE?
Thanks!